It's correct, but I recommend using a formal style for a proof by induction. For example:

Let $P(n)$ be the proposition that $T(n) = n\log_2(n)$, where $n = 2^k$ and $k$ is any positive integer,
i.e. (in terms of $k$ alone) that $T\left(2^k\right) = 2^k\!\cdot\! k$, where $k$ is any positive integer.